In order to prove Theorem REF , we basically follow the strategy of the authors' previous work [1]}. This reduces the Carleman inequality (REF ) to obtaining \(L^p-L^q\)  boundedness of a Fourier multiplier operator of which singularity is on the \(d-2\)  dimensional unit sphere \(S^{d-2}\)  embedded in \({\mathbb {R}}^d\) . We dyadically decompose the multiplier near the sphere \(S^{d-2}\) , and obtain sharp bounds for the dyadic pieces. In this approach, the Fourier restriction-extension operator (see Section ) defined by \(S^{d-2}\)  naturally arises if we write the Fourier multiplier in the cylinderical coordinates in \({\mathbb {R}}^d\) . However, direct use of the bounds on the restriction-extension operator as in [1]} does not yield any sharp estimate when \(k\ge 2\) . To overcome the problem we need to convert the restriction-extension operator into the Bochner–Riesz operator of index \(-k\) , and make use of the endpoint estimates for the Bochner–Riesz operator. See Lemmas REF  and REF .
